Particle in a 1-D box for Conjugated Polyynes

Validity of the Particle in a 1-D box for Conjugated Polyynes

R. Viswanathan
June, 1997

The particle in a one-dimensional box model is a simple quantum mechanical model that can be used to predict the electronic energy levels for the pi electrons in long-chain conjugated unsaturated compounds. This exercise compares the results predicted by the particle-in-a-box model with semi-empirical quantum mechanical calculations.

According to the particle in a 1-D box model, the energy of the electron in a quantum state n is given by:

En = n²h²/8ml²

where h is the Planck's constant, m is the mass of the electron, and l is the length of polyyne. The length of the box is estimated using the C-C triple, single, and C-H bond lengths. The number of pi electrons is 4 times the number of triple bonds in the conjugated polyyne. The HOMO (n) is one half the number of pi electrons. The energy change corresponding to the HOMO-LUMO transition is given by:

DE = E_n+1 - E_n

The wavelength corresponding to the transition is calculated as:

l_max = hc/DE


Compound	No. of carbon atoms	Length of chain	l_max 1D-box	l_max MOPAC

1,3-butadiyne	4	5.87	126	196
1,3,5-hexatriyne	6	8.43	180	210
1,3,5,7-octatetrayne	8	10.99	234	192
1,3,5,7,9-decapentayne	10	13.55	288	204
1,3,5,7,9,11-dodecahexayne	12	16.11	342	220

The structures of these polyyne molecules were minimized using the molecular mechanics method, followed by a semi-empirical method. The wavelength corresponding to the HOMO-LUMO transition were then calculated using ZINDO. The values for the wavelength of maximum absorption obtained by the semi-empirical method are compared with the particle in a box results in the following graph.

Both models predict an increase in the wavelength with increasing chain length. The effect predicted by the particle-in-a-box model is more pronounced than the results obtained using MOPAC.

The end.

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